### 3.1460 $$\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=65 $\frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2}$

[Out]

-(b*d - a*e)^2/(4*e^3*(d + e*x)^4) + (2*b*(b*d - a*e))/(3*e^3*(d + e*x)^3) - b^2/(2*e^3*(d + e*x)^2)

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Rubi [A]  time = 0.0371152, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {27, 43} $\frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]

[Out]

-(b*d - a*e)^2/(4*e^3*(d + e*x)^4) + (2*b*(b*d - a*e))/(3*e^3*(d + e*x)^3) - b^2/(2*e^3*(d + e*x)^2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^5} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^5} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^5}-\frac{2 b (b d-a e)}{e^2 (d+e x)^4}+\frac{b^2}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}+\frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{b^2}{2 e^3 (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0202525, size = 55, normalized size = 0.85 $-\frac{3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]

[Out]

-(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(12*e^3*(d + e*x)^4)

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Maple [A]  time = 0.045, size = 71, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,ae-2\,bd \right ) b}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x)

[Out]

-2/3*(a*e-b*d)*b/e^3/(e*x+d)^3-1/4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^3/(e*x+d)^4-1/2*b^2/e^3/(e*x+d)^2

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Maxima [A]  time = 1.17613, size = 132, normalized size = 2.03 \begin{align*} -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(6*b^2*e^2*x^2 + b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2 + 4*(b^2*d*e + 2*a*b*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6
*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Fricas [A]  time = 1.68925, size = 201, normalized size = 3.09 \begin{align*} -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*e^2*x^2 + b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2 + 4*(b^2*d*e + 2*a*b*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6
*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 1.10241, size = 104, normalized size = 1.6 \begin{align*} - \frac{3 a^{2} e^{2} + 2 a b d e + b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (8 a b e^{2} + 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**5,x)

[Out]

-(3*a**2*e**2 + 2*a*b*d*e + b**2*d**2 + 6*b**2*e**2*x**2 + x*(8*a*b*e**2 + 4*b**2*d*e))/(12*d**4*e**3 + 48*d**
3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12*e**7*x**4)

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Giac [A]  time = 1.14012, size = 130, normalized size = 2. \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, b^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{8 \, b^{2} d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b^{2} d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac{8 \, a b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/12*(6*b^2*e^(-2)/(x*e + d)^2 - 8*b^2*d*e^(-2)/(x*e + d)^3 + 3*b^2*d^2*e^(-2)/(x*e + d)^4 + 8*a*b*e^(-1)/(x*
e + d)^3 - 6*a*b*d*e^(-1)/(x*e + d)^4 + 3*a^2/(x*e + d)^4)*e^(-1)